Definition of "regular" in Stringham's "Regular figures in n-dimensional space"

Question

I've been reading Irving Stringham's 1880 thesis, "Regular Figures in n-dimensional Space" (only 14 pages!), after it was mentioned by Coxeter in Regular Polytopes (§7.x). I'm confused about what Stringham considers "regular".

The only definition he seems to give is for a regular angle: "one all of whose boundaries of any given number of dimensions are the same in form and magnitude." It seems to be implied that the same definition applies to a "regular figure". I would express this as saying "All $j$-faces are congruent, for each $j$."

This is not the same as the usual definition of regularity, which is that the symmetry group acts transitively on flags. For instance, the deltahedra, such as the equilateral triangular bipyramid, have the property that all faces of a given rank are congruent. So do the rhombic dodecahedron and rhombic triacontahedron.

So, fine; we can just bear in mind that Stringham is using "regular" to mean something different. But on page 2, he says "the number of regular 4-fold angles that can be made up of regular polyhedral angles is five", i.e., the vertex figure must be one of the five Platonic solids.

This seems to be false under his definition of regular. My questions are:

1. Is this indeed an error, or is the defect in my understanding?
2. If it is an error, does it impact the further conclusions of the paper? Or can we just replace the definition of "regular angle" with "one whose symmetry group acts transitively on all maximal chains of mutually incident faces" and then everything follows?

Answer 1

Too long for a comment:

Stringham gave a talk about the content of his thesis here in the Seminar of Felix Klein in Göttingen on Monday, 1880/11/29, you can look at the scans here:

Ueber reguläre Körper im vier-dimensionalen Raum. It includes hand-drawn figures very similar to the one in the paper you linked and some that are not in the printed paper like this one:

It's written in German (presumably by Stringham's hand), and it gives the identical definition to the paper and does not talk about transitivity the way we do today.

The Felix Klein protocols do give an answer to the Question "Was Stringham at some point aware of the action of the symmetry groups?", because this talk was only the first in a series of three talks he gave in Göttingen, the other two being

Monday, 1881/02/11: Ueber diejenigen Gruppen von Bewegungen der dreifach ausgedehnten Kugel, in sich, welche die ihr einbeschriebenen regulären Körper zur Selbstdeckung bringen. [About those groups of movements of the 3d-sphere in itself, which bring a regular body inscribed in it to self-congruence.]

and

Montag, 1881/05/23: Ueber eine $\mu$-$\mu$-deutige Zuordnung einer Gruppe auf sich selbst. [About a $\mu$-$\mu$-guous assignment of a group to itself]

So while unsure if there's a bug in the original paper, I think he figured it out sometimes in 1881. It is not like he's not warning us when finishing his paper with

(The quotation is from a poem by Alfred Tennyson)